where \(l^{\mu }\)  is the tangent vector to the null geodesic \(\gamma \)  and \(\lambda \)  is the affine parameter. Later, it has also been shown that ANEC can be violated [1]}, [2]}, [3]}. If two points are connected by a null geodesic \(\gamma \)  parametrized by \(\lambda \) , then (REF ) can be expressed as
\(\int _{\gamma }T_{\mu \nu }l^{\mu }l^{\nu }=\int _{\gamma }T_{\mu \nu }(x)l^{\nu }(x)dx^{\mu }=\int _{\gamma }\mathcal {J}_{\mu }(x)dx^{\mu },\) 
